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<h1 id="autoregressiveModelSequenceType">AutoregressiveModelSequence</h1><p>
Represents a sequence of multivariate autoregressive (AR) models with increasing order $p$.
The AR models should be stored as <a class="groops-file" href="fileFormat_matrix.html">matrix file</a> in the <a class="groops-ref" href="fundamentals.autoregressiveModel.html">GROOPS definition of
AR models</a>.
The required AR models can be computed with <a class="groops-program" href="CovarianceMatrix2AutoregressiveModel.html">CovarianceMatrix2AutoregressiveModel</a>,
and passed to this class through
<strong class="groops-config-element">inputfileAutoregressiveModel</strong> in increasing order.</p><p>The main purpose of AutoregressiveModelSequence is to use AR models of the form
\[
  \label{eq:ar-model}
  \mathbf{y}_e(t_i) = \sum_{k=1}^p \mathbf{\Phi}^{(p)}_k\mathbf{y}_e(t_{i-k}) + \mathbf{w}(t_i),
  \hspace{5pt} \mathbf{w}(t_i) \sim \mathcal{N}(0, \mathbf{\Sigma}^{(p)}_\mathbf{w}),
\]to create pseudo-observation equations
\[
  \label{eq:pseudo-observations-transformed}
  0 = \bar{\mathbf{\Phi}} \Delta\mathbf{y} + \bar{\mathbf{w}}, \hspace{5pt} \bar{\mathbf{w}} \sim
  \mathcal{N}(0, \bar{\mathbf{\Sigma}}_{\bar{\mathbf{w}}}),
\]with
\[
  \label{eq:pseudo-observations-ar}
  \bar{\mathbf{\Phi}} =
  \begin{bmatrix}
    \mathbf{I} & & & & & \\
    -\mathbf{\Phi}^{(1)}_1 & \mathbf{I} & & & &  \\
    -\mathbf{\Phi}^{(2)}_2 & -\mathbf{\Phi}^{(2)}_1 & \mathbf{I} & & & \\
    -\mathbf{\Phi}^{(3)}_3 & -\mathbf{\Phi}^ {(3)}_2 & -\mathbf{\Phi}^ {(3)}_1 & \mathbf{I} & &  \\
    & -\mathbf{\Phi}^{(3)}_3 & -\mathbf{\Phi}^ {(3)}_2 & -\mathbf{\Phi}^ {(3)}_1 & \mathbf{I} &  \\
    & & \ddots & \ddots & \ddots & \ddots  \\
  \end{bmatrix},
  \hspace{15pt}
  \bar{\mathbf{\Sigma}}_{\bar{\mathbf{w}}} =
  \bar{\mathbf{\Sigma}}_{\bar{\mathbf{w}}} =
  \begin{bmatrix}
    \mathbf{\Sigma}^{(0)}_{\mathbf{w}} & & & & & \\
    & \mathbf{\Sigma}^{(1)}_{\mathbf{w}} & & & & \\
    & & \mathbf{\Sigma}^{(2)}_{\mathbf{w}} & & & \\
    & & & \mathbf{\Sigma}^{(3)}_{\mathbf{w}} & & \\
    & & & & \mathbf{\Sigma}^{(3)}_{\mathbf{w}} &  \\
    & & & & & \ddots \\
  \end{bmatrix}.
\]used to constrain high-frequency temporal gravity field variations (see
<a class="groops-program" href="KalmanSmootherLeastSquares.html">KalmanSmootherLeastSquares</a>, <a class="groops-program" href="NormalsBuildShortTimeStaticLongTime.html">NormalsBuildShortTimeStaticLongTime</a>,
<a class="groops-program" href="PreprocessingSst.html">PreprocessingSst</a>).</p><p>The corresponding normal equation coefficient matrix is given by
\[
  \label{eq:ar-normals}
  \bar{\mathbf{\Phi}}^T\bar{\mathbf{\Sigma}}^{-1}_{\bar{\mathbf{w}}}\bar{\mathbf{\Phi}}
\]and if all AR models are estimated from the same sample its inverse is a block-Toeplitz covariance matrix
\[
  (\mathbf{\Sigma}_{\mathbf{y}_m})_{ij} =
  \begin{cases}
 \mathbf{\Sigma}(|j-i|) & \text{for } i \leq j \\
 \mathbf{\Sigma}(|j-i|))^T & \text{otherwise}
 \end{cases},
\]which can be computed using <a class="groops-program" href="AutoregressiveModel2CovarianceMatrix.html">AutoregressiveModel2CovarianceMatrix</a>.</p><p>A detailed description with applications can be found in:
Kvas, A., Mayer-Gürr, T. GRACE gravity field recovery with background model uncertainties.
J Geod 93, 2543–2552 (2019). <a href="https://doi.org/10.1007/s00190-019-01314-1" target="_blank">https://doi.org/10.1007/s00190-019-01314-1</a>
</p>
<table class="table table-hover">
<tr class="table-primary"><th>Name</th><th>Type</th><th>Annotation</th></tr>
<tr class="table-light"><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset">autoregressiveModelSequenceType</div></div></td><td>sequence</td><td></td></tr>
<tr class=""><td class="m-0"><div class="h-100 config-tree depth-1"><div class="h-100 config mustset-unbounded">inputfileAutoregressiveModel</div></div></td><td>filename</td><td>matrix file containing an AR model</td></tr>
<tr class="table-light"><td class="m-0"><div class="h-100 config-tree depth-1"><div class="h-100 config optional">sigma0</div></div></td><td>double</td><td>a-priori sigma for white noise covariance</td></tr>
</table>

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